c ) The Law of Value
Capitalism is based on exchange : it first presents itself as "an immense accumulation of commodities."  But though it could not exist without exchange, capitalism is not merely the production of commodities; it grows and develops even by fighting against simple commodity production.  Capital is fundamentally based on a particular type of exchange, the exchange between living labour and stored labour. The difference between Marx and the classical economists lies primarily in his creation of the concept of labour power : this concept reveals the secret of surplus-value, since it differentiates between necessary-labour and surplus labour.
How do commodities confront each other ? By what mechanism can one determine that x quantity of A has the same value as y quantity of B ? Marx does not try to find the explanation for xA = yB in the concrete nature of A and B, in their respective qualities, but in a quantitative relation : A and B can only be exchanged in the proportion xA = yB because they both contain a quantity of "something common"  to both of them. If we abstract the concrete and useful nature of A and B, they retain only one thing in common : they are both "products of labour."  A and B are exchanged in proportions determined by the respective quantities of labour crystallised in them. The quantities of labour are measured by their duration. The concept of socially necessary labour time, developed by further analysis, is an abstraction : one cannot calculate what an hour of socially necessary labour represents in a given society. But the distinction between abstract and concrete labour allows Marx to understand the mechanism of exchange and to analyse a particular form of exchange : the wage system.
"The best points in my book are : 1 ) the two-fold character of labour, according to whether it is expressed in use value or exchange value. ( All understanding of the facts depends upon this. ) It is emphasised immediately, in the first chapter... 
The buying and selling of all commodities, including the labour force, obeys what Marx calls the law of value. At first that law seems to be quite simple : commodities are exchanged according to their value, which is determined by the labour time socially necessary for their production. Yet in the third volume of Capital Marx asserts that :
"The exchange of commodities at their values, or approximately at their values, requires a much lower stage than their exchange at their prices of production, which requires a definite level of capitalist development." 
In fact, the law of value is analysed as both the cause and the consequence of a long and contradictory evolution, which we will try to summarise.
Exchange appears in primitive societies only when the degree of productivity allows people to produce more than they need to satisfy their own needs. The division of labour appears, as well as money, which "serves as a universal measure of value" :  exchange-value thus seems to acquire some sort of autonomy, embodied by the money-lender and the merchant, who earn their living out of the circulation of money, and in fact derive their living from the surplus-labour of productive working people. Money implies prices : price is the monetary form of value, although it does not coincide with value. The relation between supply and demand interferes at three levels : there is competition 1 ) among the sellers, 2 ) among the buyers, 3 ) between the sellers and the buyers. The relation between supply and demand causes a fall or a rise of price below or above the value of the commodity. But in a given period of time, and within the limits of these oscillations, the value of a commodity is not determined by competition, but by its cost of production. The value of the commodity is determined by the socially necessary labour time; its price, by the relation between supply and demand. The law of value "is none other than that which, within the fluctuations of trade periods, necessarily levels out the price of a commodity to its cost of production." 
So far we have only considered the case of simple commodity production. Capitalism develops the law of value and makes the relation of price to value extremely complex. Primitive capitalist accumulation is based on :
a ) the transformation of labour-power into a commodity, which implies that labour-power freely appears on the market as an element distinct from the others in the labour process;
b ) the accumulation of a considerable capital which is to be invested in industry.
The large sums gathered under the mercantile system from the 15th to the 17th centuries were used for this purpose. In a totally different situation, one of the aims of the destruction of the kulaks and NEPmen which started in 1928, in Russia, was to allow the State to seize a considerable stock of values in order to invest them in industry. In both cases, the development of commercial capital was the step that had to precede a great industrial boom. Produced by the development of exchange, capital itself spreads exchange throughout the planet and thereby modifies, not the law of value, but the way it manifests itself : the forms of value are transformed in order to maintain and fully develop the content of the law. The distinction price/value existed before capitalism, but industrial capitalism modifies it. We know that price oscillates around value according to the fluctuation of supply and demand. But capitalist society creates a dynamic movement in the relation price/value.
"What will be the consequence of the rising price of a commodity ? A mass of capital will be thrown into that flourishing branch of industry and this influx of capital into the domain of the favoured industry will continue until it yields the ordinary profits or, rather, until the price of its products, through overproduction, sinks below the cost of production. " 
Marx analyses this problem more systematically in the third volume of Capital :
"Owing to the different organic compositions of capitals invested in different lines of production, and, hence, owing to the circumstance that -- depending on the different percentage which the variable part makes up in a total capital of a given magnitude-capitals of equal magnitude put into motion very different quantities of labour, they also appropriate very different quantities of surplus-labour or produce very different quantities of surplus-value. Accordingly, the rates of profit prevailing in the various branches of production are originally very different. These different rates of profit are equalised by competition to a single general rate of profit, which is the average of all these different rates of profit. The profit accruing in accordance with this general rate of profit to any capital of a given magnitude, whatever its organic composition, is called the average profit. The price of a commodity, which is equal to its cost-price plus the share of the annual average profit on the total capital invested ( not merely consumed ) in its production that falls to it in accordance with the conditions of turnover, is called its price of production." 
This process is the equalisation of the rate of profit : the development of exchange gives birth to a market price, which oscillates with the fluctuations of competition within the limits we have described. The movement of market prices seems to negate the law of value. But the circulation of capital, its never ending search for branches where the cost of production is as low as possible, tends to make all rates of profit uniform. Capitalism tends to create what Marx called a "capitalist communism"  where all surplus-value is redistributed. A price of production is created as a sort of average of the oscillations of the market prices of each commodity.
"The price thus equalised, which divides up the social surplus value equally among the individual capitals in proportion to their sizes, is the price of production of commodities, the centre around which the oscillation of the market prices moves." 
Like market prices, the price of production seems to be a new negation of the law of value, since the price of commodities is composed of their cost of production plus the average profit :
"It would seem, therefore, that here the theory of value is incompatible with the actual process, incompatible with the real phenomena of production..." 
We must nevertheless think of society as a whole and consider the entire capitalist process of production.
"The capital invested in some spheres of production has a mean, or average, composition, that is, it has the same, or almost the same composition as the average social capital." 
In other sectors, it does not coincide with value; a phenomenon of "compensation" appears.
"The assumption that the commodities of the various spheres of production are sold at their value merely implies, of course, that their value is the centre of gravity around which their prices fluctuate, and their continual rises and drops tend to equalise. There is also the market-value... to be distinguished from the individual value of particular commodities produced by different producers. The individual value of some of these commodities will be below their market-value ( that is, less labour-time is required for their production than expressed in the market-value ) while that of others will exceed the market-value." 
The merit of Marx's analysis is that he tries to show a direct link between the supply-demand relation and the question of labour time ( as he did by distinguishing between value and price ).
"For a commodity to be sold at its market-value, i.e., proportionally to the necessary social labour contained in it, the total quantity of social labour used in producing the total mass of this commodity must correspond to the quantity of the social want for it, i.e., the effective social want. Competition, the fluctuations of market-prices which correspond to the fluctuations of demand and supply, tend continually to reduce to this scale the total quantity of labour devoted to each kind of commodity.". 
There is no contradiction between value, on the one hand, and the cost of production plus the average profit on the other. The operation of capitalism, by transforming surplus-value into profit, itself establishes the proportion of the value of a commodity which represents the cost of production and the portion which represents the average profit. The average profit, though it appears as "something outside,"  is merely the product of the total capital invested by society.
"It is evident that from the point of view of the total social capital the value of the commodities produced by it ( or, expressed in money, their price ) = value of constant capital + value of variable capital + surplus-value." 
"It is evident that the average profit can be nothing but the total mass of surplus-values allotted to the various quantities of capital proportionally to their magnitudes in the different spheres of production.". 
By its double negation of the law of value through the market price and the price of production, capitalism merely reinforces and extends the dominion of the law of value. Value now acquires a "modified" form : but the transformation of values into prices of production, and the creation of a market value distinct from the individual value, realise the law while generalising it.
"The commodities -- taken en masse and on a social scale -- are sold at their values." 
Marx sums up the process by which the law of value asserts itself through its double negation :
"What competition, first in a single sphere, achieves is a single market-value and market-price derived from the various individual values of commodities. And it is competition of capitals in different spheres, which first brings out the price of production equalising the rates of profit in the different spheres. The latter process requires a higher development of capitalist production than the previous one." 
"...This always resolves itself to one commodity receiving too little of the surplus-value while another receives too much, so that the deviations from the value which are embodied in the prices of production compensate one another. Under capitalist production, the general law acts as the prevailing tendency only in a very complicated and approximate manner, as a never ascertainable average of ceaseless fluctuations." 
These developments elucidate the historical cycle of exchange which runs its course through capitalism. "Popular" Marxism has turned the law of value into a mere regulating mechanism, disregarding what was interesting in Marx's study : the attempt to discover the dynamics of capitalism. The very movement of the law of value makes labour time one of the elements of this dynamics.
"I demonstrate that the average price of commodities can never be equal to its value precisely because the value of the commodity is determined by its labour time." 
Labour time, in fact, determines the entire social organisation of production and distribution. It regulates the proportions in which the productive forces are used for specific purposes at specific places. The law of value "asserts itself as it determines the necessary proportions of social labour, not in the general sense which applies to all societies, but only in the sense required by capitalist society; in other words, it establishes a proportional distribution of the whole social labour according to the specific needs of capitalist production" 
This is one of the reasons why capital will not be invested in a factory in India even though the production of that factory may be necessary to the survival of the population. Capital always goes where it can multiply quickly. The regulation by labour time compels capitalist society to develop a given production only where the labour time socially necessary for this production is at most equal to the average labour time. " The form in which this proportional distribution of labour asserts itself, in a state of society where the interconnection of social labour is manifested in the private exchange of the individual products of labour, is precisely the exchange value of these products." 
Such is the logic of capital : exchange-value determined by average labour time. Marx wondered if this movement itself produced the irrationality of the capitalist system. We will only deal with one aspect of the contradiction, by studying Marx's analysis of labour time.
 Marx, Capital, Vol. I ( Moscow : Progress Publishers, 1965 ), p. 35.
 See Rosa Luxemburg, The Accumulation of Capital ( New York : Monthly Review Press, 1964 ), Section Three.
 Capital, Vol. I, p. 37.
 Capital, Vol. I, p. 38.
 Marx's letter to Engels, August 24, 1867, in Marx and Engels, Selected Correspondence ( Moscow : Progress Publishers, 1965 ), p. 192.
 Capital, Vol. Ill ( Moscow : Progress Publishers, 1966 ), p. 177.
 Capital, Vol. I, p. 94.
 Marx, Wage Labor and Capital ( In Marx and Engels, Selected Works in Two Volumes, Vol. 1, Moscow : Foreign Languages Publishing House, 1962, p. 101 ).
 Wage Labor and Capital, p. 86.
 Capital, Vol. III, pp. 157-8
 Marx's letter to Engels, April 30, 1868, in Selected Correspondence, p. 206.
 Marx's letter to Engels, April 30, 1868.
 Capital, Vol. III, p. 153.
 Capital, Vol. III, p. 173.
 Capital, Vol. III, p. 178.
 Capital, Vol. III, p. 192.
 Capital, Vol. III, p. 168.
 Capital, Vol. III, p. 166.
 Capital, Vol. III, p. 174.
 Marx's letter to Engels, April 30, 1868, in Selected Correspondence, p. 207.
 Capital, Vol. III, p. 180.
 Capital, Vol. III, p. 161.
 Marx, Theorien uber den Mehrwert, in Werke, Vol. XXVI ( Berlin : Dietz Verlag, 1959 ), p. 28.
 Paul Mattick, "Value and Socialism," ( Cahiers de l'ISEA, "Serie Etudes de Marxologie," No. 9, August, 1965, pp. 140-172 ).
 Marx's Letter to Kugelmann, July 11, 1868, in Marx and Engels, Selected Correspondence, p. 209